358 XV NORMED ALGEBRAS AND SPECTRAL THEORY
Remark
(15.7.4.1) If E is a Hilbert space of finite dimension «, we can define
a scalar product (u\v) on «S?(E) = Endc(E) as in (15.7.4). As in (15.4.8) we
see that, if we put \\u\\l = (u\ u), then \\u\\ 2 is a norm on JSf(E) such that
\\uv\\2 ^ \\u\\2 • \\v\\2l it is necessarily equivalent to the usual norm \\u\\ (5.9.1),
but does not satisfy (15.1.2) unless n = 1, because ||1E||2 = n112.
(15.7.5) Let A be an arbitrary algebra with involution, and let g be a bitrace
on A. Since g(s*, s-*) = g(s, s), the left ideal n defined in (1 5.6.8) is self-
adjoint and is therefore a two-sided ideal. The vector space A/n is therefore
naturally an algebra. Since n is self-adjoint, the mapping si->s* induces an
involution on A/n such that n(s*) = (n(s))* ; also the scalar product (jc | y) on
A/n induced by g is a bitrace on A/n ; finally, if # satisfies the condition (U)
(resp. (N)), then the same is true of the scalar product on A/n.
A Hilbert algebra is defined to be an algebra A with involution, endowed
with the structure of&prehilbert space defined by a bitrace (x \ y) satisfying the
conditions (U) and (N). In other words, this scalar product satisfies the follow-
ing conditions :
(15.7.5.1) (
(15.7.5.2) (xy\z) = (y\x*z)
(15.7.5.3) For each x e A there exists a real number M^ ^ 0 such that
(15.7.5.4) The elements of A of the form xy, -with x e A and y e A, form a
total set in A.
Note that it follows from (15.7.5.1) and (15.7.5.2) that
(15.7.5.5)
For (yx \ z) = (x*y* \ z*) = (y* \ xz*) = (y \ zx*).
Furthermore, we have
0 5-7.5.6) (yx | yx) ^ Mx,(y \ y)
because (yx \ yx) = (x*y* \ x*y*) ^ Mx,(y* \ y*) =4.8) and that if w(E) = F is finite-